A260706 -id:A260706 - cp's OEIS frontend

A193832 Irregular triangle read by rows in which row n lists 2n-1 copies of 2n-1 and n copies of 2n, for n >= 1.

1, 2, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14

Offset: 1

Omar E. Pol, Aug 22 2011

Sequence of successive positive integers k in which if k is odd then k appears k times, otherwise if k is even then k appears k/2 times.
Note that an arrangement of the blocks of this sequence shows the growth of the generalized pentagonal numbers A001318 (see example).
The sums of each block give the positive integers of A129194: 1, 2, 9, 8, 25, 18, 49,...
Partial sums of A080995. - Paolo P. Lava, Aug 23 2011.
Concatenations of rows of triangles A001650 and A111650; also, seen as a flat list, the row lengths of triangle A260672 and the first differences of its row sums (cf. A260706). - Reinhard Zumkeller, Nov 17 2015
Also a(n) = number of squares in the arithmetic progression {24k + 1: 0 <= k <= n-1} [Granville]. - N. J. A. Sloane, Dec 13 2017

			a) If written as a triangle the initial rows are
  1, 2,
  3, 3, 3, 4, 4,
  5, 5, 5, 5, 5, 6, 6, 6,
  7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8,
  9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10,
  ...
Row sums give A126587.
b) An application using the blocks of this sequence: the illustration of the growth of an arrangement which represents the generalized pentagonal numbers A001318. For example; the first 9 positive initial terms: 1, 2, 5, 7, 12, 15, 22, 26, 35.
.
.         9
.       8 9
.     8 7 9
.   8 6 7 9
. 8 6 5 7 9
. 6 4 5 7 9
. 4 3 5 7 9
. 2 3 5 7 9
. 1 3 5 7 9
...

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Andrew Granville, Squares in arithmetic progressions and infinitely many primes, arXiv:1708.06951 [math.NT], 2017.
Andrew Granville, Squares in arithmetic progressions and infinitely many primes, The American Mathematical Monthly, 124.10 (2017): 951-954. See p. 952.

Cf. A001318, A080995, A126587, A129194, A221671, A221672.

Cf. A001650, A111650, A260672, A260706.

a193832 n k = a193832_tabf !! (n-1) !! (k-1)
a193832_row n = a193832_tabf !! (n-1)
a193832_tabf = zipWith (++) a001650_tabf a111650_tabl
a193832' n = a193832_list !! (n - 1)
a193832_list = concat a193832_tabf
-- Reinhard Zumkeller, Nov 15 2015

Array[Join @@ MapIndexed[ConstantArray[#, #/(1 + Boole[First@ #2 == 2])] &, {2 # - 1, 2 #}] &, 7] // Flatten (* or *)
Table[If[k <= 2 n - 1, 2 n - 1, 2 n], {n, 7}, {k, 3 n - 1}] // Flatten (* Michael De Vlieger, Dec 14 2017 *)

a(n) = sqrt(8n/3) plus or minus 1 [Granville] - N. J. A. Sloane, Dec 13 2017

If 8 <= n <= 52, then a(n-1) < a(n) if and only if n is in A221672. - Jonathan Sondow, Dec 14 2017

Edited by N. J. A. Sloane, Dec 13 2017

A260672 Table read by rows: T(n,k) = n - A001318(k), k = 0 .. A193832(n)-1.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 2, 1, 4, 3, 2, 5, 4, 3, 0, 6, 5, 4, 1, 7, 6, 5, 2, 0, 8, 7, 6, 3, 1, 9, 8, 7, 4, 2, 10, 9, 8, 5, 3, 11, 10, 9, 6, 4, 12, 11, 10, 7, 5, 0, 13, 12, 11, 8, 6, 1, 14, 13, 12, 9, 7, 2, 15, 14, 13, 10, 8, 3, 0, 16, 15, 14, 11, 9, 4, 1, 17, 16

Offset: 0

Reinhard Zumkeller, Nov 15 2015

Column k starts at row A001318(k); each column = A001477.

			.   0:    0
.   1:    1   0
.   2:    2   1   0
.   3:    3   2   1
.   4:    4   3   2
.   5:    5   4   3   0
.   6:    6   5   4   1
.   7:    7   6   5   2   0
.   8:    8   7   6   3   1
.   9:    9   8   7   4   2
.  10:   10   9   8   5   3
.  11:   11  10   9   6   4
.  12:   12  11  10   7   5   0
.  13:   13  12  11   8   6   1
.  14:   14  13  12   9   7   2
.  15:   15  14  13  10   8   3   0
.  16:   16  15  14  11   9   4   1
.  17:   17  16  15  12  10   5   2
.  18:   18  17  16  13  11   6   3
.  19:   19  18  17  14  12   7   4
.  20:   20  19  18  15  13   8   5  .

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened
Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, Doron Zeilberger, A pentagonal number sieve, J. Combin. Theory Ser. A 82 (1998), no. 2, 186-192.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Cf. A001318, A193832 (row lengths), A000041, A087960, A054440, A260664, A260706 (row sums).

a260672 n k = a260672_tabf !! n !! k
a260672_row n = a260672_tabf !! n
a260672_tabf = map (takeWhile (>= 0) . flip map a001318_list . (-)) [0..]

Number of m-tuples of partitions of n that have no part in common = Sum(A087960(k)*A000041(T(n,k))^m: k = 0 .. A193832(n+1)-1), e.g. A054440 (m=2) and A260664 (m=3); see Wilf link: p. 2, (3).

A212760 Number of (w,x,y,z) with all terms in {0,...,n}, w even, and x = y + z.

Original entry on oeis.org

1, 3, 12, 20, 45, 63, 112, 144, 225, 275, 396, 468, 637, 735, 960, 1088, 1377, 1539, 1900, 2100, 2541, 2783, 3312, 3600, 4225, 4563, 5292, 5684, 6525, 6975, 7936, 8448, 9537, 10115, 11340, 11988, 13357, 14079, 15600, 16400, 18081, 18963, 20812, 21780, 23805

Offset: 0

Clark Kimberling, May 29 2012

A signed version is A122576.
For a guide to related sequences, see A211795.
Partial sums of the positive elements of A129194. - Omar E. Pol, Dec 28 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A211795.

Cf. A001318, A260706.

a212760 = a260706 . fromInteger . a001318 . (+ 1)
-- Reinhard Zumkeller, Nov 17 2015

[(n+1)*(n+2)*(2*n+3+(-1)^n)/8 : n in [0..50]]; // Wesley Ivan Hurt, Jul 22 2014

A212760:=n->(n+1)*(n+2)*(2*n+3+(-1)^n)/8: seq(A212760(n), n=0..50); # Wesley Ivan Hurt, Jul 22 2014

t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[(Mod[w, 2] == 0) && x == y + z, s++],
{w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
Map[t[#] &, Range[0, 50]]  (* A212760 *)
Table[(n + 1) (n + 2) (2 n + 3 + (-1)^n)/8, {n, 0, 50}] (* Wesley Ivan Hurt, Jul 22 2014 *)
CoefficientList[Series[(1 + 2 x + 6 x^2 + 2 x^3 + x^4)/((1 + x)^3 (1 - x)^4), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 23 2014 *)

a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7).
G.f.: ( 1+2*x+6*x^2+2*x^3+x^4 ) / ( (1+x)^3*(1-x)^4 ).
a(n) = (n+1)*(n+2)*(2*n+3+(-1)^n)/8. - Wesley Ivan Hurt, Jul 22 2014
a(n) = A260706(A001318(n+1)). - Reinhard Zumkeller, Nov 17 2015
a(n) = Sum_{i=1..n+1} numerator(i^2/2). - Wesley Ivan Hurt, Feb 26 2017

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A193832 Irregular triangle read by rows in which row n lists 2n-1 copies of 2n-1 and n copies of 2n, for n >= 1.

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A260672 Table read by rows: T(n,k) = n - A001318(k), k = 0 .. A193832(n)-1.

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A212760 Number of (w,x,y,z) with all terms in {0,...,n}, w even, and x = y + z.

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